Wick rotation and analytic continuation

[wnvl2]

In quantum mechanics and quantum field theory, Wick rotation is often used to transform oscillatory integrals into exponentially convergent ones by replacing real time with imaginary time. The result is then analytically continued back to real time.

For example, a time-evolution amplitude contains a factor $e^{-iH\Delta t/\hbar}$, which leads to oscillatory integrals in path-integral formulations. After Wick rotation, these often become much easier to evaluate.

My question is: under what precise mathematical conditions is this procedure justified?

Is Wick rotation merely a computational trick, or does it rely on additional assumptions about analyticity, convergence and the singularities of the integrand? How can we be sure that analytic continuation back to real time recovers the original physical result?

I am particularly interested in references that explain the mathematical foundations of these methods, including examples where Wick rotation fails or requires special care.

 

[renormalize]

Is Wick rotation merely a computational trick, or does it rely on additional assumptions about analyticity, convergence and the singularities of the integrand? How can we be sure that analytic continuation back to real time recovers the original physical result?

Have you looked here: https://en.wikipedia.org/wiki/Wick_rotation and the references therein? In particular, that link states:

 

[wnvl2]

No. In the meantime, I found the answer also in Contour Deformation, Pinches, and Causal Prescriptions.

A Wick rotation is justified only if the integration contour can be rotated in the complex energy plane without changing the integral, except for any contributions that must be included.

The main things to check are whether the integrand is analytic in the region through which the contour is deformed, whether the contributions from the arcs at infinity vanish, and whether the rotated integral is well-defined.

One must also take care of the pole structure and the $i\varepsilon$ prescription. In particular, a pinch singularity can prevent the contour from being deformed at all.